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Chapter 10: Problem 32

Simplify. Write the answer with positive exponents only. $$t^{-7}$$

Short Answer

Expert verified
\(\frac{1}{t^7}\)

Step by step solution

01

Understand Negative Exponents

A negative exponent indicates that the base should be moved to the opposite part of a fraction. For example, \( t^{-n} = \frac{1}{t^n} \).
02

Rewrite with Positive Exponent

Apply the rule from Step 1 to the given expression \(t^{-7}\). This gives \(\frac{1}{t^7}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying expressions means making them easier to understand or work with. Often, this involves combining like terms, reducing fractions, or converting negative exponents to positive exponents.
When working with expressions, it's important to apply mathematical rules correctly and methodically. This ensures the simplified expression remains equivalent to the original.
In the given problem, we simplified the expression with a negative exponent, which means taking steps to rewrite it using positive exponents.
Positive Exponents
An exponent tells us how many times to multiply a base number by itself. For example, in the expression \(t^3\), the exponent is 3, and it tells us to multiply \(t \) by itself three times: \(t \cdot t \cdot t\).
Positive exponents are straightforward to work with because they indicate multiplication. Negative exponents, however, involve a different procedure.
In our problem, we had \(t^{-7}\). According to the rules of exponents, a negative exponent indicates that we should take the reciprocal of the base with the positive exponent. So, \(t^{-7}\) becomes \(\frac{1}{t^7}\), where 7 is now a positive exponent.
Fraction Rules
Working with fractions involves several important rules:
  • To simplify fractions, divide the numerator and the denominator by their greatest common divisor.
  • To add or subtract fractions, find a common denominator.
  • To multiply fractions, multiply the numerators together and the denominators together.
  • To divide fractions, multiply by the reciprocal of the divisor.
In the context of the given problem, understanding the fraction rule that a negative exponent tells us to move the base to the opposite part of a fraction is essential. This converts the negative exponent into a positive exponent.
Following this rule, \(t^{-7}\) becomes \(\frac{1}{t^7}\). This reciprocal change is crucial for simplifying expressions properly and ensuring they are easier to work with or more understandable.

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Most popular questions from this chapter

Simplify. Write the answers with positive exponents only. $$8^{2} 8^{-4}$$

Subtract the polynomials. $$\left(12 c^{2}+6 c-3\right)-\left(4 c^{2}-3 c+8\right)$$

Multiply the polynomials. $$\left(5 y^{2}-3 y-4\right)\left(y^{2}+4 y+7\right)$$

Add \(\left(-12 x^{4}-2 x^{2}+6 x\right)\) to \(\left(9 x^{4}+2 x^{3}+4 x\right)\).

Fill in the missing polynomial. $$(\quad)(z-5)=z^{2}-3 z-10$$
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